Super-integrable Calogero-type systems admit maximal number of Poisson structures

We present a general scheme for constructing the Poisson structure of super-integrable dynamical systems of which the rational Calogero-Moser system is the most interesting one. This dynamical system is 2N-dimensional with 2N - 1 first integrals and our construction yields 2N - 1 degenerate Poisson tensors that each admit 2(N - 1) Casimirs. Our results are quite generally applicable to all super-integrable systems and form an alternative to the traditional bi-Hamiltonian approach. (C) 2001 Published by Elsevier Science B.V.